Chapter II: Active Control of Graphene Plasmons
2.4 Tunable Planckian Thermal Emission
infrared spectral range [27, 44]. In the previous reflection measurements, the polarized absorption in the graphene nanoresonators could be tuned from 0%
to up to 24.5% for large carrier densities. In this work, the device displayed up to 3% total absorption when probed using our apparatus. This smaller number reflects the use of non-polarized light, the higher numerical aperture objective of the apparatus, the effect of the window of the vacuum stage, and the lower carrier densities used due to the onset of Poole-Frenkel tunnelling in the SiNx
at higher temperatures and high gate biases [27].
Figure 2.10: (a) Schematic of the experimental apparatus. The 70 µm × 70 µm graphene nanoresonator arrays are placed on a 1-µm-thick SiNx mem- brane with a 200-nm-thick gold backreflector. A gate bias is applied through the SiNx membrane between the underlying Si frame and graphene sheet.
The temperature-controlled stage contains a feedback controlled, heated sil- ver block that holds a 2-mm-thick copper sample carrier with a 100-µm-thick sapphire layer used for electrical isolation. The temperature is monitored with a thermocouple in the block, and the stage is held at a vacuum of 1 mtorr.
A 1-mm-thick potassium bromide (KBr) window is used to pass thermal ra- diation out of the stage, which is collected with a Cassegrain objective and passed into an FTIR with an MCT detector. (b) A representative SEM image of 30-nm-wide graphene nanoresonators on a 1-µm-thick SiNx membrane. (c) Source-drain resistance versus gate voltage curve of the device. The peak in the resistance occurs at the charge neutral point (CNP) of graphene.
In Fig. 2.11, the change in emissivity is obtained assuming unity emissivity at all frequencies for the black soot reference and normalizing the measured emission spectra accordingly. We investigate gate-tunable emissivity features as the nanoresonator doping and width is varied, as well as their polarization dependence. These results indicate that the intensity, width and energetic po- sition of the thermal radiation feature near 1,360 cm−1 are widely tunable, and that this feature is strongly polarized. The energy of this feature increases as the nanoresonator width is decreased and as the carrier density is increased.
These observations are consistent with previously reported absorption mea- surements performed on identical samples that showed a narrow absorption feature near 1,360 cm−1[27]. Thus, we attribute the prominent spectral fea- ture at 1,360 cm−1 to a Fabry-Perot plasmonic resonance from the patterned graphene. Specifically, the graphene plasmon resonant frequency should vary as ωp ∝ n1/4W−1/2, where n is the grpahene carrier density, and W is the resonator width. This behavior is in accord with the emission spectra, in which we observe a blue shift of the plasmonic resonance at increased doping and decreased graphene nanoresonator width. The intensity of the higher-energy peak increases with graphene carrier density, an effect that results from the increased polarizability of the resonant plasmonic modes. Finally, this feature is strongly polarization dependent, as we would expect for laterally confined graphene plasmonic resonant modes, and vanishes quickly as we rotate the po- larization of the probing radiation from 90◦ to 0◦ relative to the nanoresonator axis.
The microscopic processes which give rise to thermally excited plasmons in the nanoresonators are expected to correspond to the plasmonic loss processes
as emission is a reciprocal process of absorption in thermal equilibrium. The plasmonic loss processes are attributed to the factors that limit the electron mobility of the graphene, such as defect scattering, impurity scattering, and inelastic electron–electron and electron–phonon interactions [3, 26, 27, 41, 52]. In addition, plasmons have been shown to decay via loss channels associ- ated with the edges of graphene nanostructures and by coupling to substrate phonons [3, 52]. The resonant enhancement of emission from plasmon gen- erating processes is in competition with the blocking of interband transitions that act as thermal emitters in the undoped graphene, but are forbidden due to Pauli blocking when the sheet is doped [32, 48]. While interband transitions should occur across a wide range of frequencies, for patterned graphene areas, we find that doping the graphene allows for the resonant plasmonic modes to create an emission enhancement that outweighs the decrease in emission due to Pauli blocking. Thus, we get a net increase in emission near 1,360 cm−1. The plasmonic resonators also interact with vibrations in the SiNx substrate.
When the SiNx is heated, the plasmonic modes act as antennae to enhance the spontaneous thermal radiation from the nearby SiNx. The spontaneous emis- sion radiative rate is enhanced by the graphene nanoresonators, which modify the photonic mode density. The rate enhancement is correlated to the strong polarizability of the graphene at its plasmonic resonance that enhances the outcoupling of thermal radiation from the SiNx. In particular, the radiative rate is expected to be most strongly amplified within the mode volume of the resonant graphene plasmon, which for 40 nm resonators at 1.2 × 1013 cm−2 roughly corresponds to the area within 10 nm of the resonator. Therefore, we assign the net increase of thermal emission near 1,360 cm−1 to a combination
of thermal excitations in the graphene as well as thermal phonons in the SiNx that is out-coupled through the confined plasmonic modes in the graphene nanoresonators. In contrast to the emissivity features seen in the high-energy peak, the lower-energy emissivity modulation feature near 730 cm−1 shows an extremely weak polarization dependence and no noticeable dependence on graphene nanoresonator width. As the carrier density is increased, there is a small, non-monotonic increase in intensity for this feature, but it shows no spec- tral shift. Finally, unlike the higher-energy peak, the lower-energy peak is also observed in the bare, unpatterned graphene, where it appears as a slightly nar- rower feature. The low-energy feature is related to an optically active phonon in the SiNx substrate. This phonon mode is strongly absorbing (emitting) and is typically located near 850 cm−1. The large divergence in the SiNx permit- tivity due to this phonon, however, creates an additional λ/4nSiNx condition in the structure that leads to a destructive interference effect, resulting in an absorption (emission) maximum at 730 cm−1. When graphene is placed on top of the SiNx, the intraband and interband transitions in the graphene act to modify the surface impedance of the device. The result is that increasing the doping in the graphene leads to a stronger destructive interference effect, which manifests as larger emission from the SiNx layer. The graphene plasmons can couple to the SiNx phonons to create new surface phonon plasmon polariton modes [3, 4, 14, 52]. Since the observed low-energy emissivity features show no polarization dependence, an increase in direct emission from the SiNx layer likely plays the dominant role in creating the feature at 730 cm−1.
Figure 2.11: (a) Carrier density dependence of change in emissivity with re- spect to the CNP for 40-nm-wide graphene nanoresonators at 250◦C. (b) Width dependence of change in emissivity for 20-, 30-, 40-, 50-, 60-nm-wide nanores- onators at 250◦C and for a carrier density of 1.2 × 1013 cm−2. The black line indicates the emissivity changes of bare, unpatterned graphene at the same carrier density and temperature. (c) Polarization dependence of the emissivity change for 40 nm graphene nanoresonators at 250 ◦C for a carrier density of 1.2 × 1013 cm−2.
Here we test our structure as a mid-infrared spontaneous light source at higher speeds, and demonstrate a 2 kHZ modulation of graphene-nanoresonator- coupled thermal emission. We performed time-resolved emission measurements on 50-nm-wide resonators at 250◦C. A 2-kHz-modulated square wave signal was applied to the structure, with an “off” voltage of 0V, corresponding to the CNP of graphene and an “on” voltage of 60V, corresponding to a graphene carrier density of 1.2×1013cm−2. The emission modulation was measured as a raw voltage signal from a FTIR MCTA detector using an infrared filter with transmission peaked at 1,383 cm−1 and central bandwidth of approximately 30 cm−1. This filter was selected to match the resonance frequency of the 50nm resonators at a doping of 1.2× 1013cm−2, therefore isolating the plas- monic signal. The measurement results along with the applied voltage temporal waveform are shown in Fig. 2.12. A clearly modulated emission signal is seen in response to the input square wave. In these measurements, the maximum modulation frequency was 2 kHz due to limitations in the speed of the detec- tor and the RC time constant of the combined graphene nanoresonator device, contact resistance, and electrical leads. This frequency is not indicative of the inherent upper limits of the structure itself. As shown in Fig. 2.12, the applied voltage signal exhibits a sharp rise time, indicating that the primary limitations here are from the detector response.
Figure 2.12: Temporal waveform of applied voltage signal (black line) and detector signal of emission from 50 nm ribbons at 250◦C (green line).
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C h a p t e r 3
NON-EQUILIBRIUM GRAPHENE PLASMONS AND GAIN
In this chapter, we discuss non-equilibrium graphene plasmons excitations orig- inating from a hot carrier distribution created by ultrafast optical pumping.
Plasmon emission is a decay path of photoexcited carriers in graphene that has been theoretically proposed, but remained elusive experimentally. We present a theoretical model [17] and resulting predictions for plasmon emission as an ultrafast and ultrabright light emitting mechanism.
3.1 Carrier dynamics in graphene upon ultrafast optical excitation Carrier relaxation in graphene is now understood to occur via several stages and decay channels. The promptly excited carriers with a non-Fermi-like distri- bution undergo carrier-carrier and carrier-plasmon scatterings on a 10-fs time scale, followed by Auger recombination and optical phonon emission. Excited carriers eventually reach a complete equilibrium with the lattice and environ- ment through direct or disorder-assisted acoustic phonon emission, which occur on a picosecond timescale. These carrier relaxation processes in graphene upon optical pumping are depicted in Fig. 3.1 [4, 6, 10, 13, 26]. Several theoret- ical studies have proposed that plasmon emission is another competing decay channel [7, 24]. Previous studies have predicted and revealed the strong in- terplay of plasmons and particle/hole excitations in graphene, which plays a significant role in reducing the lifetime of photoexcited charge carrier [3, 8, 9, 18]. Experimental evidence for optically generated non-equilibrium plasmons
was provided by near-field microscopy measurements, where an increase in the Drude weight and the form of the resultant dispersion relations were consistent with graphene plasmons at an elevated carrier temperature upon ultrafast opti- cal excitation [15, 27]. Optical control of graphene plasmons allows excitation and modulation of graphene plasmons on ultrafast time scales.
Figure 3.1: Carrier relaxation processes in graphene under ultrafast optical excitation: (i) Sharply peaked distribution of photoexcited carriers upon optical pumping. (ii) Carriers with a non-Fermi-like distribution undergoing carrier- carrier scattering on a 10-fs time scale. (iii) Carriers in a quasi-equilibrium state. (iv) Carriers that have been thermalized under interband processes, but are still hotter than the lattice. (v) Complete equilibrium between the carriers and the lattice.
The ability to achieve inversion and plasmon gain is of fundamental interest, and is also interesting owing to the potential for a coherent amplification or lasing medium from the infrared to THz spectral region [14, 19–21]. A co- herent terahertz radiation was observed due to the parametric amplification of Josephson plasma waves in layered superconductors [22]. It has been theo- retically predicted that graphene plasmons can experience gain via stimulated plasmon emission in photoinverted graphene at excitation levels achievable via optical, electrical, or diffusion pumping [5, 16, 17, 23]. The only experimental evidence for graphene plasmon gain, to best of our knowledge, was provided by
polarization-dependent THz radiation, whose value exceeding the spontaneous emission limit was qualitatively attributed to plasmon gain [28].
3.2 Non-equilibrium plasmon dispersion relation calculations
Previous theoretical work showed that photoexcited carriers can create condi- tions for gain in non-equilibrium plasmon population via stimulated and sponta- neous plasmon emission processes [7, 16, 17]. The plasmon emission/absorption rates can be calculated using Fermi’s golden rule (FGR). According to FGR, the plasmon emission rate takes the following semi-analytical equation under the first-order approximation (first-order approximation is accurate as long as the loss/gain rates are much lower than the plasmon frequency):
g ≈αfc k θ(ω−vFk) pω2−(vFk)2
2K(ω, k)
∂Re((ω,k))
∂ω
ω=ωp(k) (3.1)
K(ω, k) =Z +1
−1
d up
1−u2f ~(ω+vFk u) 2
!
T µc
×f ~(ω−vFk u) 2
!
T µv
(3.2)
whereαf is the fine-structure constant,vF is the graphene Fermi velocity, and K(ω, k) is a measure for the phase space available for emission processes. As seen in Eq. (3.1), the emission/absorption rates critically depend on the ex- actness of the plasmon dispersion relations [17]. In order to accurately describe plasmons in photoinverted graphene, the complex plasmon frequency dispersion needs to be solved exactly in contrast to making the low loss approximation under which the plasmon frequency is considered as a real variable and the de- cay rate is solved perturbatively. We calculate the complex graphene plasmon dispersion relation, ω(k) = ωp(k) +i γp(k), by setting the dynamic dielectric